This is a complete list of all 20 side-quests available in South Park: The Stick of Truth. Many of the quests are simple and will reward you with friends required for the More Popular Than John Lennon achievement.
Side Quest List
Quest Name|Day Available|Obtained
Big Game Huntin’ with Jimbo|1|Buy the Hunting guide from Jimbo in his shop
Defeat ManBearPig|-|Available after completing Unfriend Al Gore
Dropping the Kids Off|2|Mr. Hankey’s house in the Sewer
Find Jesus|1|Speak to Priest Maxi outside the City Hall
Flower for a Princess|1|Speak to Kenny in Kupa Keep
Hide ‘n’ Seek|1|Speak to the kindergartners in the park
The Homeless Problem|1|Talk to the Mayor in City Hall
Magical Songs|2|Obtained from Jimmy once you gain access to the Elven Kingdom
ManBearPig|1|Talk to Al Gore, to the right of the cinema
Mongolian Beef|1|Talk to Mr. Kim in City Wok
Mr. Slave’s Package|1|Bring the vibrating package in the Post Office to Mr. Slave’s house. Required for the Inside Joke achievement!
Nazi Zombie Bounty|2|Talk to the police chief in the Police Station
Phase 1|3|Talk to the gnomes in your bedroom
Rats in the Cellar|1|Talk to Skeeter in his bar
Restoring the Balance|2|Speak to DogPoo in the Elven Kingdom
The She-Ogre|2|Speak to Stan in the ELven Kingdom
The Timmy Express|1|Obtained when you first go near a fast-travel flag
Unfriend Al Gore|-|Available after completing ManBearPig
Vulcan Around|1|Speak to Kevin Stoley in his bedroom
Wasted Cache|2|Speak to Mr. Mackey in the Community Center
Here is a great video playlist containing a guide for all quests in the list above.
Thanks to Sorumian for the playlist and videos. Enjoy!